Basic Analysis I — Real Analysis · Problem 6.1.13

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · pointwise-limit-bounded

Problem

Suppose there is a single constant

B

and a sequence of functions

f_n : S \to \mathbb{R}

that are bounded by

B

, that is

\left|f_n(x)\right| \leq B

for all

x \in S

. Suppose that

\{f_n\}_{n=1}^\infty

converges pointwise to

f : S \to \mathbb{R}

. Prove that

f

is bounded.